Theorem itg2leub 23546

Description: Any upper bound on the integrals of all simple functions 𝐺 dominated by 𝐹 is greater than (∫₂‘𝐹), the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.)

Assertion

Ref	Expression
itg2leub	⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → ((∫2‘𝐹) ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴)))

Distinct variable groups:   𝐴,𝑔   𝑔,𝐹

Proof of Theorem itg2leub

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2651	. . . . 5 ⊢ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}
2	1	itg2val 23540	. . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
3	2	adantr 480	. . 3 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → (∫2‘𝐹) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
4	3	breq1d 4695	. 2 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → ((∫2‘𝐹) ≤ 𝐴 ↔ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ≤ 𝐴))
5	1	itg2lcl 23539	. . . . 5 ⊢ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ*
6		supxrleub 12194	. . . . 5 ⊢ (({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ* ∧ 𝐴 ∈ ℝ*) → (sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑧 ≤ 𝐴))
7	5, 6	mpan 706	. . . 4 ⊢ (𝐴 ∈ ℝ* → (sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑧 ≤ 𝐴))
8	7	adantl 481	. . 3 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → (sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑧 ≤ 𝐴))
9		eqeq1 2655	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = (∫1‘𝑔) ↔ 𝑧 = (∫1‘𝑔)))
10	9	anbi2d 740	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ (𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔))))
11	10	rexbidv 3081	. . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔))))
12	11	ralab 3400	. . . 4 ⊢ (∀𝑧 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑧 ≤ 𝐴 ↔ ∀𝑧(∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴))
13		r19.23v 3052	. . . . . . 7 ⊢ (∀𝑔 ∈ dom ∫1((𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ (∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴))
14		ancomst 467	. . . . . . . . 9 ⊢ (((𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ ((𝑧 = (∫1‘𝑔) ∧ 𝑔 ∘𝑟 ≤ 𝐹) → 𝑧 ≤ 𝐴))
15		impexp 461	. . . . . . . . 9 ⊢ (((𝑧 = (∫1‘𝑔) ∧ 𝑔 ∘𝑟 ≤ 𝐹) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (∫1‘𝑔) → (𝑔 ∘𝑟 ≤ 𝐹 → 𝑧 ≤ 𝐴)))
16	14, 15	bitri 264	. . . . . . . 8 ⊢ (((𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (∫1‘𝑔) → (𝑔 ∘𝑟 ≤ 𝐹 → 𝑧 ≤ 𝐴)))
17	16	ralbii 3009	. . . . . . 7 ⊢ (∀𝑔 ∈ dom ∫1((𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ ∀𝑔 ∈ dom ∫1(𝑧 = (∫1‘𝑔) → (𝑔 ∘𝑟 ≤ 𝐹 → 𝑧 ≤ 𝐴)))
18	13, 17	bitr3i 266	. . . . . 6 ⊢ ((∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ ∀𝑔 ∈ dom ∫1(𝑧 = (∫1‘𝑔) → (𝑔 ∘𝑟 ≤ 𝐹 → 𝑧 ≤ 𝐴)))
19	18	albii 1787	. . . . 5 ⊢ (∀𝑧(∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ ∀𝑧∀𝑔 ∈ dom ∫1(𝑧 = (∫1‘𝑔) → (𝑔 ∘𝑟 ≤ 𝐹 → 𝑧 ≤ 𝐴)))
20		ralcom4 3255	. . . . . 6 ⊢ (∀𝑔 ∈ dom ∫1∀𝑧(𝑧 = (∫1‘𝑔) → (𝑔 ∘𝑟 ≤ 𝐹 → 𝑧 ≤ 𝐴)) ↔ ∀𝑧∀𝑔 ∈ dom ∫1(𝑧 = (∫1‘𝑔) → (𝑔 ∘𝑟 ≤ 𝐹 → 𝑧 ≤ 𝐴)))
21		fvex 6239	. . . . . . . 8 ⊢ (∫1‘𝑔) ∈ V
22		breq1 4688	. . . . . . . . 9 ⊢ (𝑧 = (∫1‘𝑔) → (𝑧 ≤ 𝐴 ↔ (∫1‘𝑔) ≤ 𝐴))
23	22	imbi2d 329	. . . . . . . 8 ⊢ (𝑧 = (∫1‘𝑔) → ((𝑔 ∘𝑟 ≤ 𝐹 → 𝑧 ≤ 𝐴) ↔ (𝑔 ∘𝑟 ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴)))
24	21, 23	ceqsalv 3264	. . . . . . 7 ⊢ (∀𝑧(𝑧 = (∫1‘𝑔) → (𝑔 ∘𝑟 ≤ 𝐹 → 𝑧 ≤ 𝐴)) ↔ (𝑔 ∘𝑟 ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴))
25	24	ralbii 3009	. . . . . 6 ⊢ (∀𝑔 ∈ dom ∫1∀𝑧(𝑧 = (∫1‘𝑔) → (𝑔 ∘𝑟 ≤ 𝐹 → 𝑧 ≤ 𝐴)) ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴))
26	20, 25	bitr3i 266	. . . . 5 ⊢ (∀𝑧∀𝑔 ∈ dom ∫1(𝑧 = (∫1‘𝑔) → (𝑔 ∘𝑟 ≤ 𝐹 → 𝑧 ≤ 𝐴)) ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴))
27	19, 26	bitri 264	. . . 4 ⊢ (∀𝑧(∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴))
28	12, 27	bitri 264	. . 3 ⊢ (∀𝑧 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑧 ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴))
29	8, 28	syl6bb 276	. 2 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → (sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴)))
30	4, 29	bitrd 268	1 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → ((∫2‘𝐹) ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521   = wceq 1523   ∈ wcel 2030  {cab 2637  ∀wral 2941  ∃wrex 2942   ⊆ wss 3607   class class class wbr 4685  dom cdm 5143  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ∘_𝑟 cofr 6938  supcsup 8387  ℝcr 9973  0cc0 9974  +∞cpnf 10109  ℝ^*cxr 10111   < clt 10112   ≤ cle 10113  [,]cicc 12216  ∫₁citg1 23429  ∫₂citg2 23430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xadd 11985  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-xmet 19787  df-met 19788  df-ovol 23279  df-vol 23280  df-mbf 23433  df-itg1 23434  df-itg2 23435

This theorem is referenced by:  itg2itg1  23548  itg2le  23551  itg2seq  23554  itg2lea  23556  itg2mulclem  23558  itg2splitlem  23560  itg2split  23561  itg2mono  23565  ftc1anclem5  33619